Counterparty Risk

Advanced Methods of Risk Management

Umberto Cherubini

Learning Objectives

• In this lecture you will learn

1. The difference between futures-style and Over-the-Counter markets.

2. The Credit Valuation Adjustment (CVA) of derivative transactions (linear/non linear)

3. The impact of dependence between risk of the underlying asset and default of the counterparty.

OTC vs Futures-style

• Over-the-Counter• Bilateral relationship• Customized

products• Low basis risk• Low liquidity• Relevant risk

– Market– Counterparty

• Futures-style• Organized market• Standardized

products• High basis risk • High liquidity• Relevant risks

– Market– Basis risk

Derivatives on OTC markets

• Most of financial derivative contracts, and particularly those with retail counterparties are traded on what is called Over-the-Counter (OTC) market

• The OTC market allows the construction of customized positions for hedging or investment purposes

• The cost is illiquidity and credit risk (counterparty risk)

The simplest example

• Consider a lineare OTC contract, i.e. forward, determined at time 0.

• Remember that if we only focus on the risk of price changes in the underlying asset we have

CF(t) = v(t,T)EQ[S(T) –F(0)] = S(t) – P(t,T)F(0)

where F(0) is the forward price at time 0. • Notice that the product is linear, meaning delta

= 1 and the replicating portfolio is static.

Counterparty risk

• We make the assumption that: default occurs at time T, which is the maturity of the contract: this is a simplifying assumption that will be relaxed later one.

• We assume that if at maturity the marked-to- market value of the derivative contract is positive for the counterparty which goes in default, the other party is compelled to abide by the contract and pay its obligation. On the other side, if the value of the contract is negative for the counterparty in default the other party has an exposure equal to that value, with the same degree of seniority of the other liabilities. This assumption corresponds to the reality of legal provisions of counter party risk.

Long and short positions

• The value of the impact of counter party risk requires to distinguish the sign, long or short of the position. This is because counterparty risk is triggered by two events:– Default of the counterparty– The contract is “out-of-the-money” for the party in

dafault, that it the contract has negative value for the counterparty in default.

• So, in case on the delivery date we have S(T) > F(0) the contract is in-the-money for the party long in the contract. If instead it is S(T) < F(0) the contract is in-the-money for the short party. In the former case the long party in the contract will be exposed to default risk, in the latter the short one will.

Default and loss

• Denote A the long party of the contract and B the short one.

• Let us introduce characteristic functions 1A and 1B assuming value 1 if the party A or B is in a default state and zero otherwise.

• Definiamo RRA e RRB i tassi di recupero delle due controparti. Nello stesso modo definiamo le loss-given-default LgdA = 1 – RRA e LgdB = 1 – RRB.

Risk of the long party

• The pay-off value of the forward contract must take into account both its sign and its value in caso of default of the relevant counterparty.

• From the viewpoint of the long end of the contract we have

CFA(T) = max[S(T) – F(0),0](1 –1B) +

max[S(T) – F(0),0]RRB1B – – max[F(0) –S(T),0] =

CF(T) – LgdB1Bmax[S(T) – F(0),0]

Risk of the short party

• For the short end of the contract, the default event is relevant only in the hypothesis that the contract ends in-the-money.

• From the viewpoint of the counterparty

CFB(T) = max[F(0) – S(T),0](1 –1A) +

max[F(0) – S(T),0]RRA1A –

– max[S(T) – F(0),0] =

– CF(T) – LgdA1Amax[F(0) – S(T),0]

Counterparty risk

• Counterparty risk corresponds to a short position is options.

• The option is of the call type for the long endo of the contract and of the put type for the other end of the contract.

• Exercise of the option is contingent on two events – The value of the underlying asset at time T– Default event of the relevant counterparty

Contract evaluation

• The value of the product from the point of view of the long end of the contract will be given by

CFA(t) = S(t) – v(t,T)F(0) – EQ[v(t,T)LgdB1Bmax[S(T) – F(0),0]]

• From the viewpoint of the short end of the contract we will then have

CFB(T) = – S(t) + v(t,T)F(0) – EQ[v(t,T)LgdA1Amax[F(0) – S(T),0]]

Risk factors

• Counterparty risk is represented by EQ[v(t,T)Lgdi1imax[(S(T) – F(0)),0]]with i = A, B and = 1(–1) for call (put) options

• Counterparty risk is made by – Interest rate risk– Market risk of the underlying– Credit risk of the counterparty – Recovery risk

• All these factors may be correlated.

A simple model

• In what follows we will assume that

– Interest rate is independent of the other risk factors

– Default risk of the counterparty is not dependent on other risk factors.

Evaluation

• The value of counterparty risk is then

v(t,T)EQ[Lgdi1i] EQ[max[(S(T) – F(0)),0]]• Notice tha in case of a zero-coupon-bond issued by

the party i we have

Di = v(t,T) – v(t,T)EQ[Lgdi1i],or

Di = v(t,T) – v(t,T)ELi,

with ELi = EQ[Lgdi1i] the expected loss.• In case of independence, then

ELi v(t,T) EQ[max[(S(T) – F(0)),0]

Effects of counterparty risk

• Effect 1: ruling out counterparty risk leads to undervaluation of the overall exposure to credit risk

• Effect 2: if one does not consider counterparty risk, he comes out with the wrong price, and the wrong hedge.

• Effect 3: counterparty risk makes linear product non linear, so that changes in volatility may affect the value of the contract even though it is linear and one would not expect any effect.

Greek letters

• The sensitivity of the contract to small changes in the underlying is no more that of a linear contract. We get in fact

A = 1 – ELBN()B = – 1 + ELAN(– )

with

=(ln(S(t)/F(0))+(r + ½ 2)(T – t))/[(T – t)1/2 ]

Gamma and Vega

• The second order effect of finite changes in the underlying is now given by

– n()/ [S(t)(T – t)1/2]• Changes in volatility affect the value of the

position through a vega effect

– S(t)n()/ [(T – t)1/2 ]

=(ln(S(t)/F(0))+(r – ½ 2)(T – t))/[(T – t)1/2 ]

An example

• Forward contract– Notional 1 million– Volatility 20%– Maturity 1 year

• Counterparty– Loss given default (Lgd): 100%– 1 year default probability: 5%

• Counterparty risk at the origin of the contract, for both the long and the short end of the contract: 3983

Long position

-500000

-400000

-300000

-200000

-100000

0

100000

200000

300000

400000

500000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Long position delta

0,94

0,95

0,96

0,97

0,98

0,99

1

1,01

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Counterparty risk (long)

0

5000

10000

15000

20000

25000

30000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Short position

-500000

-400000

-300000

-200000

-100000

0

100000

200000

300000

400000

500000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Short position delta

-1,01

-1

-0,99

-0,98

-0,97

-0,96

-0,95

-0,94

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Counterparty risk (short)

0

5000

10000

15000

20000

25000

30000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

Counterparty risk and vol. (long)

0

5000

10000

15000

20000

25000

30000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

0,1

0,2

0,3

0,4

Counterparty risk and vol. (short)

0

5000

10000

15000

20000

25000

30000

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5

0,1

0,2

0,3

0,4

Default before maturity

• Assume now that default may occur before maturity, for example by a time .

• The value of exposure for the long position is nowmax[S() – P( ,T)F(0), 0 ]

and for the short positionmax[P( ,T)F(0) – S(), 0 ]

• The value of exposure is given by a sequence of options that will be multiplied times the value of the default probability of the counterparty in the sub-periods.

Counterparty risk

• Partition the lifetime of the contract in a grid of dates {t1,t2,…tn}

• Denote Gj(ti) the survival probability of counterparty j = A, B beyond time ti.

• Compute

[GB(ti -1) – GB(ti) ]Call(S(ti), ti; P(ti ,T)F(0), ti )

[GA(ti -1) – GA(ti) ] Put(S(ti), ti; P(ti ,T)F(0), ti )respectively for long and short positions

• Aggregate the values obtained in this way from 1 through n.

Counterparty risk in swap contracts

• In a swap cotnract both the legs are exposed to counterparty risk.

• In the event of default of one of the two parties the other takes a loss equal to the marked to market value of the contract, equal to the net value of the cash-flows.

• Remember that the net value of the swap contract is positive for the long end of the contract if the swap rate on the day of default of the contract is greater than the rate on the origin of the contract.

Swap counterparty risk exposure

• Assume the set of dates at which swap payments are made be {t1, t2,…, tn} and default of the counterparty that receives fixed payments (B) took place between time tj-1 and tj. In this case, the loss for the party paying fixed is given by

where sr is the swap rate at time tj and k is the swap rate at the origin of the contract. Notice that this is the payoff of a payer swaption (a call option on a swap).

1-n

ji1B 0,,max,Lgd kttsrttP nji

Swap counterparty risk exposure

• Assume the set of dates at which swap payments are made be {t1, t2,…, tn} and default of the counterparty that pays fixed payments (A) took place between time tj-1 and tj. In this case, the loss for the party receiving fixed is given by

where sr is the swap rate at time tj and k is the swap rate at the origin of the contract. Notice that this is the payoff of a receiver swaption (a put option on a swap).

1-n

ji1A 0,,max,Lgd nji ttsrkttP

Credit risk: long partyVulnerable Call Swaptions: Financial Institution Paying Fixed

0

0,002

0,004

0,006

0,008

0,01

0,012

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Independence

Perfect positive dependence

Swap credit risk: Baa

Fixed-PayerCorrel. 5years 10years 15years 20years 25years 30yearsRho

0 0,0259% 0,0686% 0,0935% 0,1141% 0,1385% 0,1678%0,25 0,0536% 0,1448% 0,2178% 0,2574% 0,3056% 0,3658%

0,5 0,0813% 0,2210% 0,3420% 0,4007% 0,4726% 0,5637%0,75 0,1090% 0,2971% 0,4663% 0,5440% 0,6397% 0,7617%

1 0,1367% 0,3733% 0,5905% 0,6873% 0,8068% 0,9597%

Fixed-ReceiverCorrel. 5years 10years 15years 20years 25years 30yearsRho

0 0,0040% 0,0150% 0,0355% 0,0429% 0,0491% 0,0556%0,25 0,0030% 0,0113% 0,0266% 0,0322% 0,0368% 0,0417%

0,5 0,0020% 0,0075% 0,0177% 0,0215% 0,0245% 0,0278%0,75 0,0010% 0,0038% 0,0089% 0,0107% 0,0123% 0,0139%

1 0,0000% 0,0000% 0,0000% 0,0000% 0,0000% 0,0000%

Dependence structure

• A more general approach is to account for dependence between the two main events under consideration– Exercise of the option– Default of the counterparty

• Copula functions can be used to describe the dependence structure between the two events above.

Vulnerable digital call option

• Consider a vulnerable digital call (VDC) option paying 1 euro if S(T) > K (event A). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR.

• The payoff of this option is VDC = v(t,T)[H(A,Bc)+RR H(A,B)]

= v(t,T) [Ha – H(A,B)+RR H(A,B)]

= v(t,T)Ha – (1 – RR)H(A,B)

= DC – v(t,T) Lgd C(Ha, Ha)

Vulnerable digital put option

• Consider a vulnerable digital put (VDP) option paying 1 euro if S(T) ≤ K (event Ac). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR.

• The payoff of this option is

VDP = DP – v(t,T)(1 – RR)H(Ac,B)

= P(t,T)Ha – v(t,T)(1 – RR)H(Ac,B)

= P(t,T)Ha – v(t,T)(1 – RR)[Hb – C(Ha, Hb)]

= v(t,T)(1 – Ha) – v(t,T) Lgd [Hb – C(Ha, Hb)]

= v(t,T) – VDC – v(t,T) Lgd Hb

Vulnerable digital put call parity

• Define the expected loss EL = Lgd Hb.• If D(t,T) is a defaultable ZCB issued by the

counterparty we have D(t,T) = v(t,T)(1 – EL)

• Notice that copula duality implies a clear no-arbitrage relationshipVDC + VDP = v(t,T) – v(t,T) EL = D(t,T)

• Buying a vulnerable digital call and put option from the same counterparty is the same as buying a defaultable zero-coupon bond

Vulnerable call and put options

bb

K

b

K

b

K

b

K

K

HQCHC

dHQCLgdTtvTKtSPTKtSVP

dHQCLgdTtvTKtSC

dHQCLgdTtvdDC

dVDCTKtSVC

,1

,,,:,,:,

,1,,:,

,1,

,:,

0

Vulnerable put-call parity

0

0

0

0

,1,,,:,

,1,1,,:,

,,,,:,

,,,:,,:,

dHQCLgdTtvTtKDTKtSVC

dHQCLgdTtvELTtKvTKtSC

dHQCLgdTtPTtKvTKtSC

dHQCLgdTtvtSTKtSPtSTKtSVP

b

K

b

K

b

K

b

Credit risk mitigation

• Several techniques are used on the market to mitigate countrerparty risk. The ispiration of these techniques is the structure of futures-style, markets, based on three key principles– Margins– Evaluation (marking-to-market) and settlemen t of

profits and losses before maturity of the contract.– Compensation of profits and losses on different

positions• Risk mitigation clauses make more the computation

of CVA more involved. Unfortunately, there is not much literature on the subject.

CRM: theory

• In principle one can think of different techniques to mitigate counterparty risk

1. Margin deposit at the origin of the contract

2. Position evaluation at daily on weekly period and requirement of the payment of a collateral.

3. Netting agreement so that in case of default the net exposure between the counterparty is liquidated.

CRM: practice

• According to the so-called ISDA Agreement the credit mitigating techniques used apply netting and the Credit Annex requiring periodic marking-to-market of the exposures.

• Unfortunately, there is no evidence on the diffusion of these techniques in the market practice (for example) it seems that Goldman Sachs did not use them.

A simple example

• Assume a counterparty A with CFi positions in forward contracts i = 1, 2,…,p, with delivery prices Fi and delivery dates Ti with the same counterparty B.

• The value of each position is

CFi = [Si(t) – v(t,Ti)Fi]

where = 1 represents long positions and = – 1 short positions.

CVA with netting

• Assume that the counterparty B get into default at time . The value of the exposure at that date is equal to the pay-off of a basket option

p

iii

p

ii

FTvA

AS

1

1

,

0,max

Monte Carlo simulation

• As it is well known basket options can only be evaluated by Monte Carlo simulation.

• The idea is then to select a grid of dates {t1,t2,…tn} and for each one of these to evaluate a basket option, with strike A(ti). CVA is now computed, for each date, as

[G(ti-1) – G(ti)]Basket Option(S1, …Sp, ti; A(ti), ti)

where G(ti) is the survival probability of counterparty beyond time ti.

• Extension of the use of collateral occur according to the same lines as in the univariate exposure.